Question

6. The assets​ (in billions of​ dollars) of the four wealthiest people in a particular country are 39,35,15,14. Assume that samples of size n=2 are randomly selected with replacement from this population of four values

a. After identifying the 16 different possible samples and finding the mean of each​ sample, construct a table representing the sampling distribution of the sample mean. In the​ table, values of the sample mean that are the same have been combined.

X	Probability
39
37
35
27
26.5

(Type integers or​ fractions.)

X	Probability
25
24.5
15
14.5
14

(Type integers or​ fractions.)

b. Compare the mean of the population to the mean of the sampling distribution of the sample mean.

The mean of the​ population, ____ is, (greater than/less than/equal to) the mean of the sample​ means, ___.

​(Round to two decimal places as​ needed.)

c. Do the sample means target the value of the population​ mean? In​ general, do sample means make good estimates of population​ means? Why or why​ not?

The sample means (Do Not target/target) the population mean. In​ general, sample means (Do/ Do Not) make good estimates of population means because the mean is (an unbiased/ a biased) estimator

Answer 1

a.

X	Probability
39	P(X1 = 39, X2 = 39) = (1/4) * (1/4) = 1/16
37	P(X1 = 39, X2 = 35) + P(X1 = 35, X2 = 39) = (1/4) * (1/4) + (1/4) * (1/4) = 1/8
35	P(X1 = 35, X2 = 35) = (1/4) * (1/4) = 1/16
27	P(X1 = 39, X2 = 15) + P(X1 = 15, X2 = 39) = (1/4) * (1/4) + (1/4) * (1/4) = 1/8
26.5	P(X1 = 39, X2 = 14) + P(X1 = 14, X2 = 39) = (1/4) * (1/4) + (1/4) * (1/4) = 1/8
25	P(X1 = 35, X2 = 15) + P(X1 = 15, X2 = 35) = (1/4) * (1/4) + (1/4) * (1/4) = 1/8
24.5	P(X1 = 35, X2 = 14) + P(X1 = 14, X2 = 35) = (1/4) * (1/4) + (1/4) * (1/4) = 1/8
15	P(X1 = 15, X2 = 15) = (1/4) * (1/4) = 1/16
14.5	P(X1 = 15, X2 = 14) + P(X1 = 14, X2 = 15) = (1/4) * (1/4) + (1/4) * (1/4) = 1/8
14	P(X1 = 14, X2 = 14) = (1/4) * (1/4) = 1/16

b.

Mean of the population = (39 + 35 + 15 + 14) / 4 = 25.75

Mean of the sampling distribution of the sample mean = (1/16) * 39 + (1/8) * 37 + (1/16) * 35 + (1/8) * 27 + (1/8) * 26.5 + (1/8) * 25 + (1/8) * 24.5 + (1/16) * 15 + (1/8) * 14.5 + (1/16) * 14 = 25.75

The mean of the​ population, 25.75 is, equal to the mean of the sample​ means, 25.75.

c.

The sample means target the population mean. In​ general, sample means do make good estimates of population means because the mean is an unbiased estimator.